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Modern Physics
Homework #5
1.
Compare the angular momentum of a ground-state electron in the Bohr model of
the hydrogen atom with its value in quantum theory.
2.
How much more likely is the electron in a ground-state hydrogen atom to be at
the distance ao from the nucleus than at the distance ao /2? Than at the distance
2 ao?
3.
A 2s electron in a hydrogen atom is more likely than a 2p electron to be closer to
the nucleus than r = ao (that is between r = 0 and r = ao). Verify this by calculating
the relevant probabilities?
4.
Use energy considerations to show that the greatest distance a 1-s electron can be
from the nucleus for the hydrogen atom is 2 ao. Find the probability of finding the
1-s electron at a distance greater than 2 ao according to quantum mechanics.
5.
Calculate the location at which the radial probability density is a maximum for the
2-s state of the hydrogen atom. Then calculate the expectation value of the radial
coordinate in this state. Which answer if either is consistent with the Bohr model
prediction.
6.
Can there be solutions with E < 0 for the time-independent Schrodinger equation
for a system with zero potential? Explain your reasoning.
7.
A particle of total energy 9 Vo is incident from the –x axis on a potential given by
x 0
 8 Vo

V  0
0 x a
5 Vo
x a

Find the probability that the particle will be transmitted on through to the positive
side.
8.
For a particle in a box, show that the fractional difference in the energy between
adjacent eigenvalues is
ΔE n 2 n 1
 2
En
n
and use this formula to discuss the classical limit of the system.
9.
Prove that wave function,
  A 1 2 u e
2
u 2 /2
where u 
4
km x

is a solution
to the simple harmonic oscillator and derive its associated energy eigenvalue.